Difference between revisions of "Knopp function"
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| − | Let $a \in (0,1) | + | Let $a \in (0,1)$ $ab > 1$. Define the Knopp function $K \colon \mathbb{R} \rightarrow \mathbb{R}$ by |
$$K_{a,b}(x)=\displaystyle\sum_{k=0}^{\infty} a^k \mathrm{dist}_{\mathbb{Z}} \left( b^k x \right),$$ | $$K_{a,b}(x)=\displaystyle\sum_{k=0}^{\infty} a^k \mathrm{dist}_{\mathbb{Z}} \left( b^k x \right),$$ | ||
where $\mathrm{dist}_{\mathbb{Z}}$ denotes the [[distance to integers]] function. | where $\mathrm{dist}_{\mathbb{Z}}$ denotes the [[distance to integers]] function. | ||
Revision as of 21:30, 22 January 2016
Let $a \in (0,1)$ $ab > 1$. Define the Knopp function $K \colon \mathbb{R} \rightarrow \mathbb{R}$ by $$K_{a,b}(x)=\displaystyle\sum_{k=0}^{\infty} a^k \mathrm{dist}_{\mathbb{Z}} \left( b^k x \right),$$ where $\mathrm{dist}_{\mathbb{Z}}$ denotes the distance to integers function.
Properties
Theorem: The Knopp function $K_{a,b}$ is continuous on $\mathbb{R}$ for $a \in (0,1)$ and $ab>1$.
Proof: █
Theorem: The Knopp function $K_{a,b}$ is nowhere differentiable on $\mathbb{R}$ for $a \in (0,1)$and $ab > 1$.
Proof: █
See Also
Takagi function
van der Waerden function
